Lottery paradox

First put forth by Henry Kyburg in 1961. It tries to demonstrate that three principles of rational acceptance are logically inconsistent. These principles are:

If it is very likely that a certain conclusion is true, then it is rational to accept that conclusion.

If it is rational to accept that p is the case, and it is rational to accept that q is the case, than it is rational to accept that both p and q are the case.

It is never rational to accept propositions which you realize to be inconsistent.

And now, the paradox:

Suppose that there is a lottery in which 100,000 tickets are sold (and only one will win). The probability that any one given ticket will win is very low -- 0.00001. Therefore, by principle 1, it is rational to believe that my ticket, #1, will lose. And it is also rational to believe that ticket #2 will lose. And #3, and #4, and #5, .... and #99998, and #99,999, and #100,000. According to principle 2 above, it is rational to assume that all 100,000 tickets will lose. But as already stated, one ticket will win.

Suppose propositionp is 70% likely to be true, and proposition q is also 70% likely to be true. Each proposition, taken individually, is more likely to be true than not.

Now, imagine that you are walking to work, and since it rained last night, the road is a bit muddy. Suppose p represents whether or not you will make it to work without getting mud on your suit. And q represents whether or not a car will run through a puddle, splashing and soaking your suit.

Now, according to this Principle 2, one might assume, "Since it is probable that I will reach my workplace without getting mud on my suit, and since it is probable that I will get there without getting my suit wet, I can conclude that I will probably make it to work with a clean suit."

Of course, the probability that pANDq are true is not 70%, but the probabilities multiplied, or 49%. So, you will probably arrive at work with a wet or muddy suit, and should probably consider an alternate route.

So, even if it is 99.999% certain that an individuallottery ticket will lose, it is only ~99.998% certain that two tickets will both lose, and only ~99.997% certain that all of three tickets will lose, and so forth, with decreasing probabilities that all of an increasingly larger set of tickets will lose.

By the time you reach the 100,000th ticket, the probability that all will lose is some infinitesimal fraction very close to zero, which represents the probability that a large meteor will destroy the earth before the lottery is drawn.